Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

Abstract

Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steadystate depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.

Brownian molecules formed by delayed harmonic interactions

“…Properties of the Green's function (77) as well as those corresponding to a power-law memory are discussed in detail in Ref. [76].…”

Symmetry and collective fluctuations: large deviations and scaling in population processes

“…In the Markoffian case one would evaluate the required ensemble average by substituting equation 3into the above expressions and using properties of the Dirac delta to exchange the derivative of the stochastic variable x(t) for that of the fixed variable x. However the time non-local form of equation 3implies that this substitution cannot be used to evaluate the required averages [43]. Performing this procedure leads to a non-closed FPE [52], that is a partial differential equation where the first time derivative of ( )…”

“…We show these differences by analyzing three examples: a DLE with a single τ-delay process and a GLE with an exponential memory and one with a power law memory, respectively, of the form the one-parameter Mittag-Leffler function of index υ [62,63]. While the Green's function for the DLE process and for the GLE with power law memory have also a parameter region with unstable dynamics where ( ) l t grows without bounds, all three cases possess a monotonic and a (damped) oscillatory regime [43].…”

“…Fourier characteristics and the time dependence of one-time averages of these three non-Markoffian Langevin equations have been recently studied by the present authors in[43]. The respective Green's functions are…”

Fokker–Planck description for a linear delayed Langevin equation with additive Gaussian noise

“…Here we present a solution to this problem by building on recent studies of some of the present authors [42,43]. We construct and solve the 2-time FPE including the case where the initial history is non-zero (zero initial history makes a DLE a special case of a GLE).…”